discrete_mathematics_with_applications/chapter_3/test_yourself.md

Test Yourself

Page 141

1. If P(x) is a predicate with domain D, the truth set of P(x) is denoted _______. We read these symbols out loud as _______.

\{x \in D | P(x)\}; "the set of all x in D such that P(x)."

2. Some ways to express the symbol \forall in words are _______.

for every for all, for any, for each, for arbitrary, given any

3. Some ways to express the symbol \exists in words are _______.

there exists, there exist, there exists at least one, for some, for at least one, we can find a

4. A statement of the form \forall x \in D, Q(x) is true if, and only if, Q(x) is _______ for _______.

true; every x in D.

5. A statement of the form \exists x \in D such that Q(x) is true if, and only if, Q(x) is _______ for _______.

true; at least one x in D.

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Test Yourself

Page 152

1. A negation for "All R have property $S$" is "There is _______ R that _______."

exists at least one; does not have property S.

2. A negation for "Some R have property $S$" is "_______."

"No R have property S."

3. A negation for "For every x, if x has property P then x has property $Q$" is "_______."

"There exists at least one x such that x has property P and x does not have property Q."

4. The converse of "For every x, if x has property P then x has property $Q$" is "_______."

"For every x, if x has property Q then x has property P."

5. The contrapositive of "For every x, if x has property P then x has property $Q$" is "_______."

"For every x, if x does not have property Q, then x does not have property P."

6. The inverse of "For every x, if x has property P then x has property $Q$" is "_______."

"For every x, if x does not have property P, then x does not have property Q."

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Test Yourself

Page 165

1. To establish the truth of a statement of the form "\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x)," you imagine that someone has given you an element x from D but that you have no control over what that element is. Then you need to find _______ with the property that the x the person gave you together with the _______ you subsequently found satisfy _______.

y \in E; y; P(x, y)

2. To establish the truth of a statement of the form "\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)," you need to find _______ so that no matter what _______ a person might subsequently give you, _______ will be true.

x \in D; y \in E; P(x, y)

3. Consider the statement "\forall x, \exists y \text{ such that } P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}." A negation for this statement is "_______."

"\exists x such that \forall y, the property P(x, y) is false."

4. Consider the statement "\exists x \text{ such that } \forall y, P(x, y), \text{ a property involving } x \text{ and } y, \text{ is true}." A negation for this statement is "_______."

"\forall x, \exists y such that the property P(x, y) is false."

5. Suppose P(x, y) is some property involving x and y, and suppose the statement "$\forall x \text{ in } D, \exists y \text{ in } E \text{ such that } P(x, y)$" is true. Then the statement "$\exists x \text{ in } D \text{ such that } \forall y \text{ in } E, P(x, y)$"

a. is true.

b. is false.

c. may be true or may be false.

c is the answer, it may be true or false depending on the nature of the property involving x, and y. In other words, it relies on what P(x, y) states.

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Test Yourself

Page 179

1. The rule of universal instantiation says that if some property is true for _______ in a domain, then it is true for _______.

all elements; any particular element in the domain

2. If the first two premises of universal modus ponens are written as "If x makes P(x) true, then x makes Q(x) true" and "For a particular value of a _______ , " then the conclusion can be written as "______. "

P(a) is true, Q(a) is true

3. If the first two premises of universal modus tollens are written as "If x makes P(x) true, then x makes Q(x) true" and "For a particular value of a _______ ," then the conclusion can be written as " _______. "

Q(a) is false; P(a) is false

4. If the first two premises of universal transitivity are written as "Any x that makes P(x) true makes Q(x) true" and "Any x that makes Q(x) true makes R(x) true," then the conclusion can be written as "_______."

"Any x that makes P(x) true makes R(x) true"

5. Diagrams can be helpful in testing an argument for validity. However, if some possible configurations of the premises are not drawn, a person could conclude that an argument was _______ when it was actually _______.

valid; invalid